Question

Which of the following inequalities is an algebraic expression for the shaded part of the number line above?

• A. $\left|x\right|\le 3$
• B. $\left|x\right|\le 5$
• C. $|x-2|\le 3$
• D. $|x-1|\le 4$
• E. $|x+1|\le 4$

Answer 1

The correct option is E $|x+1|\le 4$

In the given number line it follows that $-5\le x\le 3$

(a) $\left|x\right|\le 3\Rightarrow -3\le x\le 3\ne -5\le x\le 3$

Therefore, the inequality $\left|x\right|\le 3$ is not an algebraic expression for the shaded part of the given number line.

(b) $\left|x\right|\le 5\Rightarrow -5\le x\le 5\ne -5\le x\le 3$

Therefore, the inequality $\left|x\right|\le 5$ is not an algebraic expression for the shaded part of the given number line.

(c) $|x-2|\le 3\Rightarrow -3\le x-2\le 3\Rightarrow -3+2\le x-2+2\le 3+2\Rightarrow -1\le x\le 5\ne -5\le x\le 3$

Therefore, the inequality $|x-2|\le 3$ is not an algebraic expression for the shaded part of the given number line.

(d) $|x-1|\le 4\Rightarrow -4\le x-1\le 4\Rightarrow -4+1\le x-1+1\le 4+1\Rightarrow -3\le x\le 5\ne -5\le x\le 3$

Therefore, the inequality $|x-1|\le 4$ is not an algebraic expression for the shaded part of the given number line.

(e) $|x+1|\le 4\Rightarrow -4\le x+1\le 4\Rightarrow -4-1\le x+1-1\le 4-1\Rightarrow -5\le x\le 3=-5\le x\le 3$

Hence, the inequality $|x+1|\le 4$ is an algebraic expression for the shaded part of the given number line.